nLab Darboux's theorem

Idea

Darboux’s theorem states that a smooth manifold $X$ equipped with a differential 1-form $\theta$ which is sufficiently non-degenerate admits local coordinate charts $\phi_i \colon\mathbb{R}^{2n} \to X$ on which $\theta$ takes the canonical form $\phi_i^\ast \theta = \sum_{k} x^{2k} \mathbf{d} x^{2k+1}$.

In particular

• for $(X, \omega)$ a symplectic manifold there are local charts in which the symplectic form $\omega$ takes the form $\phi_i^\ast \omega = \sum_{k} \mathbf{d} x^{2k} \wedge \mathbf{d} x^{2k+1}$;

• similarly, for contact manifolds there are adapated local charts.

(e.g. Arnold 78, p. 362)

For the case of symplectic manifolds the Darboux theorem may also be read as saying that a G-structure for $G = Sp(2n)$ the symplectic group (hence an almost symplectic structure) is an integrable G-structure already when it is first-order integrable, i.e. torsion-free, i.e. symplectic. See at integrability of G-structures – Examples – Symplectic structure.

Related concepts

• canonical coordinates, canonical momenta

• integrable G-structure

References

Lecture notes include

• Andreas Čap, section 1.8 of Differential Geometry 2, 2011/2012 (pdf)

• Federica Pasquotto, Linear $G$-structures by example (pdf)

Textbook accounts include

• Vladimir Arnol'd, Mathematical methods of classical mechanics, Graduate texts in Mathematics 60 (1978)

See also

• Wikipedia, Darboux theorem

A version for supergeometry is due to

• Bertram Kostant, in Lecture Notes in Mathematics vol.570, 177, (Bleuler, K. and Reetz, A.eds), Proc. Conf. on Diff. Geom. Meth. in Math. Phys., Bonn 1975., Springer-Verlag, Berlin, 1977

A version for prequantized symplectic manifolds (prequantum bundles) is discussed in

• Frédéric Faure, Masato Tsujii, around prop. 2.15 of Prequantum transfer operator for symplectic Anosov diffeomorphism (arXiv:1206.0282)

A version for shifted symplectic structures (BV-brackets) on stacks is discussed in

• Christopher Brav, Vittoria Bussi, Dominic Joyce, A ‘Darboux theorem’ for derived schemes with shifted symplectic structure (arXiv:1305.6302)

• Oren Ben-Bassat, Christopher Brav, Vittoria Bussi, Dominic Joyce, A ‘Darboux Theorem’ for shifted symplectic structures on derived Artin stacks, with applications (arXiv:1312.0090)